Outer-convex domination in graphs

2019 ◽  
Vol 12 (01) ◽  
pp. 2050008 ◽  
Author(s):  
Jonecis A. Dayap ◽  
Enrico L. Enriquez
Keyword(s):  
Convex Set ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Convex Domination Number ◽  
Domination In Graphs

Let [Formula: see text] be a connected simple graph. A set [Formula: see text] of vertices of a graph [Formula: see text] is an outer-convex dominating set if every vertex not in [Formula: see text] is adjacent to some vertex in [Formula: see text] and [Formula: see text] is a convex set. The outer-convex domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of an outer-convex dominating set of [Formula: see text]. An outer-convex dominating set of cardinality [Formula: see text] will be called a [Formula: see text]-[Formula: see text]. In this paper, we initiate the study and characterize the outer-convex dominating sets in the join of the two graphs.

Pitchfork domination in graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050025
Author(s):  
Manal N. Al-Harere ◽  
Mohammed A. Abdlhusein
Keyword(s):  
Maximum Degree ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
New Model ◽  
Domination In Graphs

In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pitchfork dominating sets in [Formula: see text]. In this work, pitchfork domination when [Formula: see text] and [Formula: see text] is studied. Some bounds on [Formula: see text] related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph [Formula: see text] without isolated vertices have a pitchfork domination or not?

Outer-paired domination in graphs

2020 ◽  
Vol 12 (06) ◽  
pp. 2050072
Author(s):  
A. Mahmoodi ◽  
L. Asgharsharghi
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Sharp Bounds ◽  
Paired Domination Number ◽  
Vertex Set ◽  
Paired Domination ◽  
Domination In Graphs

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. An outer-paired dominating set [Formula: see text] of a graph [Formula: see text] is a dominating set such that the subgraph induced by [Formula: see text] has a perfect matching. The outer-paired domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of an outer-paired dominating set of [Formula: see text]. In this paper, we study the outer-paired domination number of graphs and present some sharp bounds concerning the invariant. Also, we characterize all the trees with [Formula: see text].

scholarly journals ON THE ROOTS OF TOTAL DOMINATION POLYNOMIAL OF GRAPHS, II

2019 ◽  
pp. 659
Author(s):  
Saeid Alikhani ◽  
Nasrin Jafari
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Domination Polynomial

Let $G = (V, E)$ be a simple graph of order $n$. A  total dominating set of $G$ is a subset $D$ of $V$, such that every vertex of $V$ is adjacent to at least one vertex in  $D$. The total domination number of $G$ is  minimum cardinality of  total dominating set in $G$ and is denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. In this paper, we study roots of the total domination polynomial of some graphs.  We show that  all roots of $D_t(G, x)$ lie in the circle with center $(-1, 0)$ and radius $\sqrt[\delta]{2^n-1}$, where $\delta$ is the minimum degree of $G$. As a consequence, we prove that if $\delta\geq \frac{2n}{3}$,  then every integer root of $D_t(G, x)$ lies in the set $\{-3,-2,-1,0\}$.

scholarly journals New results on connected dominating structures in graphs

2019 ◽  
Vol 11 (1) ◽  
pp. 52-64
Author(s):  
Libin Chacko Samuel ◽  
Mayamma Joseph
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Forbidden Subgraph ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Complete Subgraph ◽  
Strong Product ◽  
Necessary And Sufficient ◽  
Domination In Graphs ◽  
Product Of Graphs

Abstract A set of vertices in a graph is a dominating set if every vertex not in the set is adjacent to at least one vertex in the set. A dominating structure is a subgraph induced by the dominating set. Connected domination is a type of domination where the dominating structure is connected. Clique domination is a type of domination in graphs where the dominating structure is a complete subgraph. The clique domination number of a graph G denoted by γk(G) is the minimum cardinality among all the clique dominating sets of G. We present few properties of graphs admitting dominating cliques along with bounds on clique domination number in terms of order and size of the graph. A necessary and sufficient condition for the existence of dominating clique in strong product of graphs is presented. A forbidden subgraph condition necessary to imply the existence of a connected dominating set of size four also is found.

scholarly journals Neighborhood Connected k-Fair Domination Under Some Binary Operations

2019 ◽  
Vol 12 (3) ◽  
pp. 1337-1349
Author(s):  
Wardah Masanggila Bent-Usman ◽  
Rowena Isla ◽  
Sergio Canoy
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Dominating Sets ◽  
Induced Subgraph ◽  
Sharp Bounds ◽  
Binary Operations ◽  
Domination In Graphs

Let G=(V(G),E(G)) be a simple graph. A neighborhood connected k-fair dominating set (nckfd-set) is a dominating set S subset V(G) such that |N(u)  intersection S|=k for every u is an element of V(G)\S and the induced subgraph of S is connected. In this paper, we introduce and invistigate the notion of neighborhood connected k-fair domination in graphs. We also characterize such dominating sets in the join, corona, lexicographic and cartesians products of graphs and determine the exact value or sharp bounds of their corresponding neighborhood connected k-fair domination number.

scholarly journals Further Results on Dual Domination in Graphs

2020 ◽  
Vol 9 (3) ◽  
pp. 1949-1950
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Graph Parameters ◽  
Domination In Graphs

Let G = (V, E) be a simple graph. A set S V(G) is a dual dominating set of G (or bi-dominating set of G) if S is a dominating set of G and every vertex in S dominates exactly two vertices in V-S. The dual-domination number γdu(G) (or bi-domination number G bi  ) of a graph G is the minimum cardinality of the minimal dual dominating set (or dual dominating set). In this paper dual domination number and relation with other graph parameters are determined.

scholarly journals Total Partial Domination in Graphs Under Some Binary Operations

2019 ◽  
Vol 12 (4) ◽  
pp. 1643-1655
Author(s):  
Roselainie Dimasindil Macapodi ◽  
Rowena Isla
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Dominating Sets ◽  
Cartesian Products ◽  
Sharp Bounds ◽  
Binary Operations ◽  
Domination In Graphs

Let G = (V (G), E(G)) be a simple graph and let α ∈ (0, 1]. A set S ⊆ V (G) isan α-partial dominating set in G if |N[S]| ≥ α |V (G)|. The smallest cardinality of an α-partialdominating set in G is called the α-partial domination number of G, denoted by ∂α(G). An α-partial dominating set S ⊆ V (G) is a total α-partial dominating set in G if every vertex in S isadjacent to some vertex in S. The total α-partial domination number of G, denoted by ∂T α(G), isthe smallest cardinality of a total α-partial dominating set in G. In this paper, we characterize thetotal partial dominating sets in the join, corona, lexicographic and Cartesian products of graphsand determine the exact values or sharp bounds of the corresponding total partial dominationnumber of these graphs.

Doubly connected bi-domination in graphs

2020 ◽  
pp. 2150009 ◽  
Author(s):  
Mohammed A. Abdlhusein
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Finite Graph ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Isolated Vertex ◽  
Domination In Graphs

Let [Formula: see text] be a finite graph, simple, undirected and has no isolated vertex. A dominating subset [Formula: see text] of [Formula: see text] is said a bi-dominating set, if every vertex of it dominates two vertices of [Formula: see text]. The bi-domination number of [Formula: see text], denoted by [Formula: see text] is the minimum cardinality over all bi-dominating sets in [Formula: see text]. In this paper, a certain modified bi-domination parameter called doubly connected bi-domination and its inverse are introduced. Several bounds and properties are studied here. These modified dominations are applied and evaluated for several well-known graphs and complement graphs.

Connected hop domination in graphs under some binary operations

2018 ◽  
Vol 11 (05) ◽  
pp. 1850075
Author(s):  
Yamilita M. Pabilona ◽  
Helen M. Rara
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Lexicographic Product ◽  
Dominating Sets ◽  
Induced Subgraph ◽  
Binary Operations ◽  
Network Application ◽  
Domination In Graphs ◽  
Product Of Graphs

Let [Formula: see text] be a simple graph. A hop dominating set [Formula: see text] is called a connected hop dominating set of [Formula: see text] if the induced subgraph [Formula: see text] of [Formula: see text] is connected. The smallest cardinality of a connected hop dominating set of [Formula: see text], denoted by [Formula: see text], is called the connected hop domination number of [Formula: see text]. In this paper, we characterize the connected hop dominating sets in the join, corona and lexicographic product of graphs and determine the corresponding connected hop domination number of these graphs. The study of these concepts is motivated with a social network application.

scholarly journals On k-Fair Total Domination in Graphs

2021 ◽  
Vol 14 (2) ◽  
pp. 578-589
Author(s):  
Wardah Masanggila Bent-Usman ◽  
Rowena T. Isla
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Cartesian Product ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Domination In Graphs ◽  
Product Of Graphs

Let G = (V (G), E(G)) be a simple non-empty graph. For an integer k ≥ 1, a k-fairtotal dominating set (kf td-set) is a total dominating set S ⊆ V (G) such that |NG(u) ∩ S| = k for every u ∈ V (G)\S. The k-fair total domination number of G, denoted by γkf td(G), is the minimum cardinality of a kf td-set. A k-fair total dominating set of cardinality γkf td(G) is called a minimum k-fair total dominating set or a γkf td-set. We investigate the notion of k-fair total domination in this paper. We also characterize the k-fair total dominating sets in the join, corona, lexicographic product and Cartesian product of graphs and determine the exact values or sharpbounds of their corresponding k-fair total domination number.

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